It will need to be repotted this Spring, wait till it gets warmer and the sunlight is stronger, unless you live in the south where it is already Spring, If you use commercial potting soil, add extra perlite and maybe some grani grit for good measure ( though the perlite will suffice). Note the white (beige) mineral scale on the pot- this means the soil is old and has accumulated mineral salts building up.

How much and how often are you watering?

krdress, I don't grow these, but suspect it is a combination of needing repotting, previous care, light issues and the plants natural growth habit.

yep, it is P. selloum (P.bipinnatifidum).

it needs to be staked. those thick roots are aerial that anchor the plant , so they often grow down into soil, but not always. do not cut them off.

it needs more then 1 cup a week for sure: it likes to be moist, do not let it dry up more then 1/3 of pot height. it also needs to be fertilized.

pot size it totally fine, no need to put it in a larger pot.

it looks leggy: it likes some partial dappled sun indoors for half a day to do well. eastern or western window is best.

When you re-pot you can plant the brown "stem" beneath the soil level. If you want to keep it in the same clay pot you probably can. May have to do a little root trimming. Also, if you do want to use same pot I would try to get the salt build up off. Soak whole pot overnight it water then scrub it off.